Sen. H. John Heinz III (R-Penn.) asked President Carter yesterday for a "full and speedy inquiry" into the Justice Department's handling of an investigation of a series of overdrafts by Budget Director Bert Lance on a bank where he was chairman.

The overdrafts, totaling $150,000, were used partly to help finance his campaign for the 1974 gubernatorial nomination in Georgia. The overdrafts were allegedly repaid with interest.

In March 1976, the Comptroller of the Currency referred its file on the overdrafts to the Justice Department for possible prosecution. The legal question was whether Lance misused the funds of the Calhoun (Ga.) First National Bank. In January 1975, Lance became president of the National Bank of Georgia.

In his letter Heinz questioned the timing of the termination of the case by then U.S. Attorney John W. Stokes in Atlanta.

Stokes, who closed out the case one day before Lnace was nominated as budget director, said the termination was his own decision because there was no case. He said he did not act under any pressure.

Richard L. Thornburgh, who was an assistant attorney general at the time, has corroborated Stokes' view that the case was "not prosecutable."

Heinz, in his letter, said there are three questions that must be answered:

"First and foremost, we must know whether the U.S. Attorney's decision was an independent one, supported by the staff working on the case and by the evidence they had collected.

"Secondly, it should be made clear whether the U.S. Attorney or his staff was subject to outside pressure from any quarter, and if so, how such pressure was handled."

"Finally," Heinz wrote, "the inquiry should be attempt to determine, if at all possible, whether any undue, special consideration was given to Mr. Lance in this case either because of his special associations or because of the position of high authority he was slated to hold."